X-ray computed tomography (CT) has found extensive clinical applications in cancer, heart, and brain imaging. As CT has been increasingly used for cancer screening and pediatric imaging, there has arisen a push to reduce the radiation dose of clinical CT scans to become as low as reasonably achievable. Thus, iterative image reconstruction has been playing a more significant role in CT imaging. Iterative image reconstruction algorithms, as compared with traditional analytical algorithms, are promising in reducing the radiation dose while improving the CT image quality.
In X-ray computed tomography (CT), iterative reconstruction can be used to generate images. While there are various iterative reconstruction (IR) methods, such as the algebraic reconstruction technique, one common IR method is optimizing the expression
      argmin    x    ⁢          ⁢      {                                                  x            -            ℓ                                    W        2            +              β        ⁢                                  ⁢                  U          ⁡                      (            x            )                                }  to obtain the argument x that minimize the expression. For example, in X-ray CT A is the system matrix that represents X-ray trajectories (i.e., line integrals) along various rays from a source through an object OBJ to an X-ray detector (e.g., the X-ray transform corresponding to projections through the three-dimensional object OBJ onto a two-dimensional projection image l), l represents projection images taken at a series of projection angles and corresponding to the log-transform of the measured X-ray intensity at the X-ray detector, and x represents the reconstructed image of the X-ray attenuation of the object OBJ. The notation ∥g∥W2 signifies a weighted inner product of the form 0.5×gT Wg, wherein W is the weight matrix. For example, the weight matrix W can weigh the pixel values according to their noise statistics (e.g., the signal-to-noise ratio), in which case the weight matrix W is diagonal when the noise of each pixel is statistically independent. The data-matching term ∥Ax−l∥W2 is minimized when the forward projection A of the reconstructed image x provides a good approximation to all measured projection images l. In the above expression, U(x) is a regularization term, and β is a regularization parameter that weights the relative contributions of the data-matching term and the regularization term.
IR methods augmented with regularization can have several advantages over other reconstruction methods such as filtered back-projection. For example, IR methods augmented with regularization can produce high-quality reconstructions for few-view projection data and in the presence of significant noise. For few-view, limited-angle, and noisy projection scenarios, the application of regularization operators between reconstruction iterations seeks to tune the final and/or intermediate results to some a priori model. For example, enforcing positivity for the attenuation coefficients can provide a level of regularization based on the practical assumption that there are no regions in the object OBJ that cause an increase (i.e., gain) in the intensity of the X-ray radiation.
Other regularization terms can similarly rely on a priori knowledge of characteristics or constraints imposed on the reconstructed image. For example, minimizing the “total variation” (TV) in conjunction with projection on convex sets (POCS) is also a very popular regularization scheme. The TV-minimization algorithm assumes that the image is predominantly uniform over large regions with sharp transitions at the boundaries of the uniform regions, which is generally true for an image of a discrete number of organs, each with an approximately constant X-ray absorption coefficient (e.g., bone having a first absorption coefficient, the lungs having second coefficient, and the heart having a third coefficient). When the a priori model corresponds well to the image object OBJ, these regularized IR algorithms can produce impressive images even though the reconstruction problem is significantly underdetermined (e.g., few-view scenarios), missing projection angles, or noisy.
While the above formulation of the IR method, which uses post-log projection data (i.e., projection data that has been converted from intensity to attenuation by calculating the logarithm of the intensity measurements), can generate better quality images at low dose than filtered-back-projection methods, a continued push to reduce radiation dosage to patients creates pressures and incentives to provide even better image reconstruction at lower X-ray dosages. Tomographic image reconstruction for low-dose CT is increasingly challenging as dose continues to be reduced in clinical applications, and, due to electronic noise, data may contain negative values for which logarithm is undefined. Pre-log methods and post-log methods have been separately proposed to improve various aspects of CT image reconstruction, and each type of method has its own advantages and disadvantages. For example, pre-log methods have the disadvantage of slow convergence due to the nonlinear transformation from image to measurement, but pre-log methods also have the advantage that, in theory, they can improve image quality for low-dose data by accurately modeling the noise. On the other hand, the post-log methods have the advantage of fast convergence, but a disadvantage that, for low-count CT data, image quality can be relatively poor due to noise amplification in the logarithm calculation used to convert the projection data from intensity to attenuation.